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Addressing the problem of consistent statistical-function estimation for two classes (GACS and SC) of nonstationary processes, this book is organized for readers with different prerequisites. The first three chapters are intended for readers to grasp the main ideas, and contain results in the form of theorems with sketches of proofs and illustrative examples. The last chapters present two-part mathematical proofs: the first part consists of formal manipulations, aimed at advanced readers such as engineering graduate students; the second consists of justification of the formal manipulations, for specialists such as mathematicians.

Dedication iii

Acknowledgements xiii

Introduction xv

1 Background 1

1.1 Second-Order Characterization of Stochastic Processes 1

1.1.1 Time-Domain Characterization 1

1.1.2 Spectral-Domain Characterization 2

1.1.3 Time-Frequency Characterization 4

1.1.4 Wide-Sense Stationary Processes 5

1.1.5 Evolutionary Spectral Analysis 5

1.1.6 Discrete-Time Processes 7

1.1.7 Linear Time-Variant Transformations 8

1.2 Almost-Periodic Functions 10

1.2.1 Uniformly Almost-Periodic Functions 11

1.2.2 AP Functions in the Sense of Stepanov,Weyl, and Besicovitch 12

1.2.3 Weakly AP Functions in the Sense of Eberlein 13

1.2.4 Pseudo AP Functions 14

1.2.5 AP Functions in the Sense of Hartman and Ryll-Nardzewski 15

1.2.6 AP Functions Defined on Groups and with Values in Banach and Hilbert Spaces 16